Question: If 40$\blacklozenge$ represents a three-digit positive integer with a ones digit of $\blacklozenge$ and 1$\blacklozenge$ is a two-digit positive integer with a ones digit of $\blacklozenge$, what value of $\blacklozenge$ makes the equation 40$\blacklozenge$ $\div$ 27 = 1$\blacklozenge$ true?
Explanation: To make our task easier, we rewrite $40\blacklozenge$ and $1\blacklozenge$ as $400+\blacklozenge$ and $10+\blacklozenge,$ respectively. Now we can setup up an equation to solve for $\blacklozenge$: \[\frac{400+\blacklozenge}{27}=10+\blacklozenge.\] Multiplying both sides by 27 gives \[ 400 + \blacklozenge = 27 ( 10 + \blacklozenge).\] Expanding the right-hand sides gives \[ 400 + \blacklozenge = 270 + 27 \cdot \blacklozenge.\] Subtracting 270 and $\blacklozenge$ from both sides gives \[130 = 26 \cdot \blacklozenge,\] and dividing by 26 gives $\blacklozenge = \boxed{5}.$